For $x>1$, is there a linear transform which transforms $\frac1x$ to $\frac{1-x}{2}$ for the hypergeometric function $_2F_1$, i.e., $_2F_1(a,b;c;\frac1x)$ in terms of $_2F_1(\alpha,\beta;\gamma;\frac{1-x}{2})$ ?
Linear transform from $_2F_1(a,b;c;\frac1x)$ to $_2F_1(\alpha,\beta;\gamma;\frac{1-x}{2})$?
1
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calculus
special-functions
linear-transformations
hypergeometric-function
integral-transforms
1 Answers
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Unless we put some restrictions on parameters $a$, $b$, $c$, the argument of the hypergeometric function $_2F_1(a,b;c;x)$ can only transform between six cross-ratios $$x,\quad 1-x,\quad \frac1x,\quad \frac{x}{x-1}, \quad \frac{1}{1-x},\quad \frac{x-1}{x},$$ hence the answer to your question : there isn't.